// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Hauke Heibel <hauke.heibel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_TRANSFORM_H
#define EIGEN_TRANSFORM_H

namespace Eigen {

namespace internal {

    template <typename Transform> struct transform_traits
    {
        enum
        {
            Dim = Transform::Dim,
            HDim = Transform::HDim,
            Mode = Transform::Mode,
            IsProjective = (int(Mode) == int(Projective))
        };
    };

    template <typename TransformType,
              typename MatrixType,
              int Case =
                  transform_traits<TransformType>::IsProjective ? 0 : int(MatrixType::RowsAtCompileTime) == int(transform_traits<TransformType>::HDim) ? 1 : 2,
              int RhsCols = MatrixType::ColsAtCompileTime>
    struct transform_right_product_impl;

    template <typename Other, int Mode, int Options, int Dim, int HDim, int OtherRows = Other::RowsAtCompileTime, int OtherCols = Other::ColsAtCompileTime>
    struct transform_left_product_impl;

    template <typename Lhs, typename Rhs, bool AnyProjective = transform_traits<Lhs>::IsProjective || transform_traits<Rhs>::IsProjective>
    struct transform_transform_product_impl;

    template <typename Other, int Mode, int Options, int Dim, int HDim, int OtherRows = Other::RowsAtCompileTime, int OtherCols = Other::ColsAtCompileTime>
    struct transform_construct_from_matrix;

    template <typename TransformType> struct transform_take_affine_part;

    template <typename _Scalar, int _Dim, int _Mode, int _Options> struct traits<Transform<_Scalar, _Dim, _Mode, _Options>>
    {
        typedef _Scalar Scalar;
        typedef Eigen::Index StorageIndex;
        typedef Dense StorageKind;
        enum
        {
            Dim1 = _Dim == Dynamic ? _Dim : _Dim + 1,
            RowsAtCompileTime = _Mode == Projective ? Dim1 : _Dim,
            ColsAtCompileTime = Dim1,
            MaxRowsAtCompileTime = RowsAtCompileTime,
            MaxColsAtCompileTime = ColsAtCompileTime,
            Flags = 0
        };
    };

    template <int Mode> struct transform_make_affine;

}  // end namespace internal

/** \geometry_module \ingroup Geometry_Module
  *
  * \class Transform
  *
  * \brief Represents an homogeneous transformation in a N dimensional space
  *
  * \tparam _Scalar the scalar type, i.e., the type of the coefficients
  * \tparam _Dim the dimension of the space
  * \tparam _Mode the type of the transformation. Can be:
  *              - #Affine: the transformation is stored as a (Dim+1)^2 matrix,
  *                         where the last row is assumed to be [0 ... 0 1].
  *              - #AffineCompact: the transformation is stored as a (Dim)x(Dim+1) matrix.
  *              - #Projective: the transformation is stored as a (Dim+1)^2 matrix
  *                             without any assumption.
  *              - #Isometry: same as #Affine with the additional assumption that
  *                           the linear part represents a rotation. This assumption is exploited
  *                           to speed up some functions such as inverse() and rotation().
  * \tparam _Options has the same meaning as in class Matrix. It allows to specify DontAlign and/or RowMajor.
  *                  These Options are passed directly to the underlying matrix type.
  *
  * The homography is internally represented and stored by a matrix which
  * is available through the matrix() method. To understand the behavior of
  * this class you have to think a Transform object as its internal
  * matrix representation. The chosen convention is right multiply:
  *
  * \code v' = T * v \endcode
  *
  * Therefore, an affine transformation matrix M is shaped like this:
  *
  * \f$ \left( \begin{array}{cc}
  * linear & translation\\
  * 0 ... 0 & 1
  * \end{array} \right) \f$
  *
  * Note that for a projective transformation the last row can be anything,
  * and then the interpretation of different parts might be slightly different.
  *
  * However, unlike a plain matrix, the Transform class provides many features
  * simplifying both its assembly and usage. In particular, it can be composed
  * with any other transformations (Transform,Translation,RotationBase,DiagonalMatrix)
  * and can be directly used to transform implicit homogeneous vectors. All these
  * operations are handled via the operator*. For the composition of transformations,
  * its principle consists to first convert the right/left hand sides of the product
  * to a compatible (Dim+1)^2 matrix and then perform a pure matrix product.
  * Of course, internally, operator* tries to perform the minimal number of operations
  * according to the nature of each terms. Likewise, when applying the transform
  * to points, the latters are automatically promoted to homogeneous vectors
  * before doing the matrix product. The conventions to homogeneous representations
  * are performed as follow:
  *
  * \b Translation t (Dim)x(1):
  * \f$ \left( \begin{array}{cc}
  * I & t \\
  * 0\,...\,0 & 1
  * \end{array} \right) \f$
  *
  * \b Rotation R (Dim)x(Dim):
  * \f$ \left( \begin{array}{cc}
  * R & 0\\
  * 0\,...\,0 & 1
  * \end{array} \right) \f$
  *<!--
  * \b Linear \b Matrix L (Dim)x(Dim):
  * \f$ \left( \begin{array}{cc}
  * L & 0\\
  * 0\,...\,0 & 1
  * \end{array} \right) \f$
  *
  * \b Affine \b Matrix A (Dim)x(Dim+1):
  * \f$ \left( \begin{array}{c}
  * A\\
  * 0\,...\,0\,1
  * \end{array} \right) \f$
  *-->
  * \b Scaling \b DiagonalMatrix S (Dim)x(Dim):
  * \f$ \left( \begin{array}{cc}
  * S & 0\\
  * 0\,...\,0 & 1
  * \end{array} \right) \f$
  *
  * \b Column \b point v (Dim)x(1):
  * \f$ \left( \begin{array}{c}
  * v\\
  * 1
  * \end{array} \right) \f$
  *
  * \b Set \b of \b column \b points V1...Vn (Dim)x(n):
  * \f$ \left( \begin{array}{ccc}
  * v_1 & ... & v_n\\
  * 1 & ... & 1
  * \end{array} \right) \f$
  *
  * The concatenation of a Transform object with any kind of other transformation
  * always returns a Transform object.
  *
  * A little exception to the "as pure matrix product" rule is the case of the
  * transformation of non homogeneous vectors by an affine transformation. In
  * that case the last matrix row can be ignored, and the product returns non
  * homogeneous vectors.
  *
  * Since, for instance, a Dim x Dim matrix is interpreted as a linear transformation,
  * it is not possible to directly transform Dim vectors stored in a Dim x Dim matrix.
  * The solution is either to use a Dim x Dynamic matrix or explicitly request a
  * vector transformation by making the vector homogeneous:
  * \code
  * m' = T * m.colwise().homogeneous();
  * \endcode
  * Note that there is zero overhead.
  *
  * Conversion methods from/to Qt's QMatrix and QTransform are available if the
  * preprocessor token EIGEN_QT_SUPPORT is defined.
  *
  * This class can be extended with the help of the plugin mechanism described on the page
  * \ref TopicCustomizing_Plugins by defining the preprocessor symbol \c EIGEN_TRANSFORM_PLUGIN.
  *
  * \sa class Matrix, class Quaternion
  */
template <typename _Scalar, int _Dim, int _Mode, int _Options> class Transform
{
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar, _Dim == Dynamic ? Dynamic : (_Dim + 1) * (_Dim + 1))
    enum
    {
        Mode = _Mode,
        Options = _Options,
        Dim = _Dim,       ///< space dimension in which the transformation holds
        HDim = _Dim + 1,  ///< size of a respective homogeneous vector
        Rows = int(Mode) == (AffineCompact) ? Dim : HDim
    };
    /** the scalar type of the coefficients */
    typedef _Scalar Scalar;
    typedef Eigen::Index StorageIndex;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3
    /** type of the matrix used to represent the transformation */
    typedef typename internal::make_proper_matrix_type<Scalar, Rows, HDim, Options>::type MatrixType;
    /** constified MatrixType */
    typedef const MatrixType ConstMatrixType;
    /** type of the matrix used to represent the linear part of the transformation */
    typedef Matrix<Scalar, Dim, Dim, Options> LinearMatrixType;
    /** type of read/write reference to the linear part of the transformation */
    typedef Block<MatrixType, Dim, Dim, int(Mode) == (AffineCompact) && (int(Options) & RowMajor) == 0> LinearPart;
    /** type of read reference to the linear part of the transformation */
    typedef const Block<ConstMatrixType, Dim, Dim, int(Mode) == (AffineCompact) && (int(Options) & RowMajor) == 0> ConstLinearPart;
    /** type of read/write reference to the affine part of the transformation */
    typedef typename internal::conditional<int(Mode) == int(AffineCompact), MatrixType&, Block<MatrixType, Dim, HDim>>::type AffinePart;
    /** type of read reference to the affine part of the transformation */
    typedef typename internal::conditional<int(Mode) == int(AffineCompact), const MatrixType&, const Block<const MatrixType, Dim, HDim>>::type ConstAffinePart;
    /** type of a vector */
    typedef Matrix<Scalar, Dim, 1> VectorType;
    /** type of a read/write reference to the translation part of the rotation */
    typedef Block<MatrixType, Dim, 1, !(internal::traits<MatrixType>::Flags & RowMajorBit)> TranslationPart;
    /** type of a read reference to the translation part of the rotation */
    typedef const Block<ConstMatrixType, Dim, 1, !(internal::traits<MatrixType>::Flags & RowMajorBit)> ConstTranslationPart;
    /** corresponding translation type */
    typedef Translation<Scalar, Dim> TranslationType;

    // this intermediate enum is needed to avoid an ICE with gcc 3.4 and 4.0
    enum
    {
        TransformTimeDiagonalMode = ((Mode == int(Isometry)) ? Affine : int(Mode))
    };
    /** The return type of the product between a diagonal matrix and a transform */
    typedef Transform<Scalar, Dim, TransformTimeDiagonalMode> TransformTimeDiagonalReturnType;

protected:
    MatrixType m_matrix;

public:
    /** Default constructor without initialization of the meaningful coefficients.
    * If Mode==Affine or Mode==Isometry, then the last row is set to [0 ... 0 1] */
    EIGEN_DEVICE_FUNC inline Transform()
    {
        check_template_params();
        internal::transform_make_affine<(int(Mode) == Affine || int(Mode) == Isometry) ? Affine : AffineCompact>::run(m_matrix);
    }

    EIGEN_DEVICE_FUNC inline explicit Transform(const TranslationType& t)
    {
        check_template_params();
        *this = t;
    }
    EIGEN_DEVICE_FUNC inline explicit Transform(const UniformScaling<Scalar>& s)
    {
        check_template_params();
        *this = s;
    }
    template <typename Derived> EIGEN_DEVICE_FUNC inline explicit Transform(const RotationBase<Derived, Dim>& r)
    {
        check_template_params();
        *this = r;
    }

    typedef internal::transform_take_affine_part<Transform> take_affine_part;

    /** Constructs and initializes a transformation from a Dim^2 or a (Dim+1)^2 matrix. */
    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline explicit Transform(const EigenBase<OtherDerived>& other)
    {
        EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename OtherDerived::Scalar>::value),
                            YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY);

        check_template_params();
        internal::transform_construct_from_matrix<OtherDerived, Mode, Options, Dim, HDim>::run(this, other.derived());
    }

    /** Set \c *this from a Dim^2 or (Dim+1)^2 matrix. */
    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& operator=(const EigenBase<OtherDerived>& other)
    {
        EIGEN_STATIC_ASSERT((internal::is_same<Scalar, typename OtherDerived::Scalar>::value),
                            YOU_MIXED_DIFFERENT_NUMERIC_TYPES__YOU_NEED_TO_USE_THE_CAST_METHOD_OF_MATRIXBASE_TO_CAST_NUMERIC_TYPES_EXPLICITLY);

        internal::transform_construct_from_matrix<OtherDerived, Mode, Options, Dim, HDim>::run(this, other.derived());
        return *this;
    }

    template <int OtherOptions> EIGEN_DEVICE_FUNC inline Transform(const Transform<Scalar, Dim, Mode, OtherOptions>& other)
    {
        check_template_params();
        // only the options change, we can directly copy the matrices
        m_matrix = other.matrix();
    }

    template <int OtherMode, int OtherOptions> EIGEN_DEVICE_FUNC inline Transform(const Transform<Scalar, Dim, OtherMode, OtherOptions>& other)
    {
        check_template_params();
        // prevent conversions as:
        // Affine | AffineCompact | Isometry = Projective
        EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(OtherMode == int(Projective), Mode == int(Projective)), YOU_PERFORMED_AN_INVALID_TRANSFORMATION_CONVERSION)

        // prevent conversions as:
        // Isometry = Affine | AffineCompact
        EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(OtherMode == int(Affine) || OtherMode == int(AffineCompact), Mode != int(Isometry)),
                            YOU_PERFORMED_AN_INVALID_TRANSFORMATION_CONVERSION)

        enum
        {
            ModeIsAffineCompact = Mode == int(AffineCompact),
            OtherModeIsAffineCompact = OtherMode == int(AffineCompact)
        };

        if (EIGEN_CONST_CONDITIONAL(ModeIsAffineCompact == OtherModeIsAffineCompact))
        {
            // We need the block expression because the code is compiled for all
            // combinations of transformations and will trigger a compile time error
            // if one tries to assign the matrices directly
            m_matrix.template block<Dim, Dim + 1>(0, 0) = other.matrix().template block<Dim, Dim + 1>(0, 0);
            makeAffine();
        }
        else if (EIGEN_CONST_CONDITIONAL(OtherModeIsAffineCompact))
        {
            typedef typename Transform<Scalar, Dim, OtherMode, OtherOptions>::MatrixType OtherMatrixType;
            internal::transform_construct_from_matrix<OtherMatrixType, Mode, Options, Dim, HDim>::run(this, other.matrix());
        }
        else
        {
            // here we know that Mode == AffineCompact and OtherMode != AffineCompact.
            // if OtherMode were Projective, the static assert above would already have caught it.
            // So the only possibility is that OtherMode == Affine
            linear() = other.linear();
            translation() = other.translation();
        }
    }

    template <typename OtherDerived> EIGEN_DEVICE_FUNC Transform(const ReturnByValue<OtherDerived>& other)
    {
        check_template_params();
        other.evalTo(*this);
    }

    template <typename OtherDerived> EIGEN_DEVICE_FUNC Transform& operator=(const ReturnByValue<OtherDerived>& other)
    {
        other.evalTo(*this);
        return *this;
    }

#ifdef EIGEN_QT_SUPPORT
    inline Transform(const QMatrix& other);
    inline Transform& operator=(const QMatrix& other);
    inline QMatrix toQMatrix(void) const;
    inline Transform(const QTransform& other);
    inline Transform& operator=(const QTransform& other);
    inline QTransform toQTransform(void) const;
#endif

    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return int(Mode) == int(Projective) ? m_matrix.cols() : (m_matrix.cols() - 1); }
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }

    /** shortcut for m_matrix(row,col);
    * \sa MatrixBase::operator(Index,Index) const */
    EIGEN_DEVICE_FUNC inline Scalar operator()(Index row, Index col) const { return m_matrix(row, col); }
    /** shortcut for m_matrix(row,col);
    * \sa MatrixBase::operator(Index,Index) */
    EIGEN_DEVICE_FUNC inline Scalar& operator()(Index row, Index col) { return m_matrix(row, col); }

    /** \returns a read-only expression of the transformation matrix */
    EIGEN_DEVICE_FUNC inline const MatrixType& matrix() const { return m_matrix; }
    /** \returns a writable expression of the transformation matrix */
    EIGEN_DEVICE_FUNC inline MatrixType& matrix() { return m_matrix; }

    /** \returns a read-only expression of the linear part of the transformation */
    EIGEN_DEVICE_FUNC inline ConstLinearPart linear() const { return ConstLinearPart(m_matrix, 0, 0); }
    /** \returns a writable expression of the linear part of the transformation */
    EIGEN_DEVICE_FUNC inline LinearPart linear() { return LinearPart(m_matrix, 0, 0); }

    /** \returns a read-only expression of the Dim x HDim affine part of the transformation */
    EIGEN_DEVICE_FUNC inline ConstAffinePart affine() const { return take_affine_part::run(m_matrix); }
    /** \returns a writable expression of the Dim x HDim affine part of the transformation */
    EIGEN_DEVICE_FUNC inline AffinePart affine() { return take_affine_part::run(m_matrix); }

    /** \returns a read-only expression of the translation vector of the transformation */
    EIGEN_DEVICE_FUNC inline ConstTranslationPart translation() const { return ConstTranslationPart(m_matrix, 0, Dim); }
    /** \returns a writable expression of the translation vector of the transformation */
    EIGEN_DEVICE_FUNC inline TranslationPart translation() { return TranslationPart(m_matrix, 0, Dim); }

    /** \returns an expression of the product between the transform \c *this and a matrix expression \a other.
    *
    * The right-hand-side \a other can be either:
    * \li an homogeneous vector of size Dim+1,
    * \li a set of homogeneous vectors of size Dim+1 x N,
    * \li a transformation matrix of size Dim+1 x Dim+1.
    *
    * Moreover, if \c *this represents an affine transformation (i.e., Mode!=Projective), then \a other can also be:
    * \li a point of size Dim (computes: \code this->linear() * other + this->translation()\endcode),
    * \li a set of N points as a Dim x N matrix (computes: \code (this->linear() * other).colwise() + this->translation()\endcode),
    *
    * In all cases, the return type is a matrix or vector of same sizes as the right-hand-side \a other.
    *
    * If you want to interpret \a other as a linear or affine transformation, then first convert it to a Transform<> type,
    * or do your own cooking.
    *
    * Finally, if you want to apply Affine transformations to vectors, then explicitly apply the linear part only:
    * \code
    * Affine3f A;
    * Vector3f v1, v2;
    * v2 = A.linear() * v1;
    * \endcode
    *
    */
    // note: this function is defined here because some compilers cannot find the respective declaration
    template <typename OtherDerived>
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const typename internal::transform_right_product_impl<Transform, OtherDerived>::ResultType
    operator*(const EigenBase<OtherDerived>& other) const
    {
        return internal::transform_right_product_impl<Transform, OtherDerived>::run(*this, other.derived());
    }

    /** \returns the product expression of a transformation matrix \a a times a transform \a b
    *
    * The left hand side \a other can be either:
    * \li a linear transformation matrix of size Dim x Dim,
    * \li an affine transformation matrix of size Dim x Dim+1,
    * \li a general transformation matrix of size Dim+1 x Dim+1.
    */
    template <typename OtherDerived>
    friend EIGEN_DEVICE_FUNC inline const typename internal::transform_left_product_impl<OtherDerived, Mode, Options, _Dim, _Dim + 1>::ResultType
    operator*(const EigenBase<OtherDerived>& a, const Transform& b)
    {
        return internal::transform_left_product_impl<OtherDerived, Mode, Options, Dim, HDim>::run(a.derived(), b);
    }

    /** \returns The product expression of a transform \a a times a diagonal matrix \a b
    *
    * The rhs diagonal matrix is interpreted as an affine scaling transformation. The
    * product results in a Transform of the same type (mode) as the lhs only if the lhs
    * mode is no isometry. In that case, the returned transform is an affinity.
    */
    template <typename DiagonalDerived> EIGEN_DEVICE_FUNC inline const TransformTimeDiagonalReturnType operator*(const DiagonalBase<DiagonalDerived>& b) const
    {
        TransformTimeDiagonalReturnType res(*this);
        res.linearExt() *= b;
        return res;
    }

    /** \returns The product expression of a diagonal matrix \a a times a transform \a b
    *
    * The lhs diagonal matrix is interpreted as an affine scaling transformation. The
    * product results in a Transform of the same type (mode) as the lhs only if the lhs
    * mode is no isometry. In that case, the returned transform is an affinity.
    */
    template <typename DiagonalDerived>
    EIGEN_DEVICE_FUNC friend inline TransformTimeDiagonalReturnType operator*(const DiagonalBase<DiagonalDerived>& a, const Transform& b)
    {
        TransformTimeDiagonalReturnType res;
        res.linear().noalias() = a * b.linear();
        res.translation().noalias() = a * b.translation();
        if (EIGEN_CONST_CONDITIONAL(Mode != int(AffineCompact)))
            res.matrix().row(Dim) = b.matrix().row(Dim);
        return res;
    }

    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& operator*=(const EigenBase<OtherDerived>& other) { return *this = *this * other; }

    /** Concatenates two transformations */
    EIGEN_DEVICE_FUNC inline const Transform operator*(const Transform& other) const
    {
        return internal::transform_transform_product_impl<Transform, Transform>::run(*this, other);
    }

#if EIGEN_COMP_ICC
private:
    // this intermediate structure permits to workaround a bug in ICC 11:
    //   error: template instantiation resulted in unexpected function type of "Eigen::Transform<double, 3, 32, 0>
    //             (const Eigen::Transform<double, 3, 2, 0> &) const"
    //  (the meaning of a name may have changed since the template declaration -- the type of the template is:
    // "Eigen::internal::transform_transform_product_impl<Eigen::Transform<double, 3, 32, 0>,
    //     Eigen::Transform<double, 3, Mode, Options>, <expression>>::ResultType (const Eigen::Transform<double, 3, Mode, Options> &) const")
    //
    template <int OtherMode, int OtherOptions> struct icc_11_workaround
    {
        typedef internal::transform_transform_product_impl<Transform, Transform<Scalar, Dim, OtherMode, OtherOptions>> ProductType;
        typedef typename ProductType::ResultType ResultType;
    };

public:
    /** Concatenates two different transformations */
    template <int OtherMode, int OtherOptions>
    inline typename icc_11_workaround<OtherMode, OtherOptions>::ResultType operator*(const Transform<Scalar, Dim, OtherMode, OtherOptions>& other) const
    {
        typedef typename icc_11_workaround<OtherMode, OtherOptions>::ProductType ProductType;
        return ProductType::run(*this, other);
    }
#else
    /** Concatenates two different transformations */
    template <int OtherMode, int OtherOptions>
    EIGEN_DEVICE_FUNC inline typename internal::transform_transform_product_impl<Transform, Transform<Scalar, Dim, OtherMode, OtherOptions>>::ResultType
    operator*(const Transform<Scalar, Dim, OtherMode, OtherOptions>& other) const
    {
        return internal::transform_transform_product_impl<Transform, Transform<Scalar, Dim, OtherMode, OtherOptions>>::run(*this, other);
    }
#endif

    /** \sa MatrixBase::setIdentity() */
    EIGEN_DEVICE_FUNC void setIdentity() { m_matrix.setIdentity(); }

    /**
   * \brief Returns an identity transformation.
   * \todo In the future this function should be returning a Transform expression.
   */
    EIGEN_DEVICE_FUNC static const Transform Identity() { return Transform(MatrixType::Identity()); }

    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& scale(const MatrixBase<OtherDerived>& other);

    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& prescale(const MatrixBase<OtherDerived>& other);

    EIGEN_DEVICE_FUNC inline Transform& scale(const Scalar& s);
    EIGEN_DEVICE_FUNC inline Transform& prescale(const Scalar& s);

    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& translate(const MatrixBase<OtherDerived>& other);

    template <typename OtherDerived> EIGEN_DEVICE_FUNC inline Transform& pretranslate(const MatrixBase<OtherDerived>& other);

    template <typename RotationType> EIGEN_DEVICE_FUNC inline Transform& rotate(const RotationType& rotation);

    template <typename RotationType> EIGEN_DEVICE_FUNC inline Transform& prerotate(const RotationType& rotation);

    EIGEN_DEVICE_FUNC Transform& shear(const Scalar& sx, const Scalar& sy);
    EIGEN_DEVICE_FUNC Transform& preshear(const Scalar& sx, const Scalar& sy);

    EIGEN_DEVICE_FUNC inline Transform& operator=(const TranslationType& t);

    EIGEN_DEVICE_FUNC
    inline Transform& operator*=(const TranslationType& t) { return translate(t.vector()); }

    EIGEN_DEVICE_FUNC inline Transform operator*(const TranslationType& t) const;

    EIGEN_DEVICE_FUNC
    inline Transform& operator=(const UniformScaling<Scalar>& t);

    EIGEN_DEVICE_FUNC
    inline Transform& operator*=(const UniformScaling<Scalar>& s) { return scale(s.factor()); }

    EIGEN_DEVICE_FUNC
    inline TransformTimeDiagonalReturnType operator*(const UniformScaling<Scalar>& s) const
    {
        TransformTimeDiagonalReturnType res = *this;
        res.scale(s.factor());
        return res;
    }

    EIGEN_DEVICE_FUNC
    inline Transform& operator*=(const DiagonalMatrix<Scalar, Dim>& s)
    {
        linearExt() *= s;
        return *this;
    }

    template <typename Derived> EIGEN_DEVICE_FUNC inline Transform& operator=(const RotationBase<Derived, Dim>& r);
    template <typename Derived> EIGEN_DEVICE_FUNC inline Transform& operator*=(const RotationBase<Derived, Dim>& r) { return rotate(r.toRotationMatrix()); }
    template <typename Derived> EIGEN_DEVICE_FUNC inline Transform operator*(const RotationBase<Derived, Dim>& r) const;

    typedef typename internal::conditional<int(Mode) == Isometry, ConstLinearPart, const LinearMatrixType>::type RotationReturnType;
    EIGEN_DEVICE_FUNC RotationReturnType rotation() const;

    template <typename RotationMatrixType, typename ScalingMatrixType>
    EIGEN_DEVICE_FUNC void computeRotationScaling(RotationMatrixType* rotation, ScalingMatrixType* scaling) const;
    template <typename ScalingMatrixType, typename RotationMatrixType>
    EIGEN_DEVICE_FUNC void computeScalingRotation(ScalingMatrixType* scaling, RotationMatrixType* rotation) const;

    template <typename PositionDerived, typename OrientationType, typename ScaleDerived>
    EIGEN_DEVICE_FUNC Transform&
    fromPositionOrientationScale(const MatrixBase<PositionDerived>& position, const OrientationType& orientation, const MatrixBase<ScaleDerived>& scale);

    EIGEN_DEVICE_FUNC
    inline Transform inverse(TransformTraits traits = (TransformTraits)Mode) const;

    /** \returns a const pointer to the column major internal matrix */
    EIGEN_DEVICE_FUNC const Scalar* data() const { return m_matrix.data(); }
    /** \returns a non-const pointer to the column major internal matrix */
    EIGEN_DEVICE_FUNC Scalar* data() { return m_matrix.data(); }

    /** \returns \c *this with scalar type casted to \a NewScalarType
    *
    * Note that if \a NewScalarType is equal to the current scalar type of \c *this
    * then this function smartly returns a const reference to \c *this.
    */
    template <typename NewScalarType>
    EIGEN_DEVICE_FUNC inline typename internal::cast_return_type<Transform, Transform<NewScalarType, Dim, Mode, Options>>::type cast() const
    {
        return typename internal::cast_return_type<Transform, Transform<NewScalarType, Dim, Mode, Options>>::type(*this);
    }

    /** Copy constructor with scalar type conversion */
    template <typename OtherScalarType> EIGEN_DEVICE_FUNC inline explicit Transform(const Transform<OtherScalarType, Dim, Mode, Options>& other)
    {
        check_template_params();
        m_matrix = other.matrix().template cast<Scalar>();
    }

    /** \returns \c true if \c *this is approximately equal to \a other, within the precision
    * determined by \a prec.
    *
    * \sa MatrixBase::isApprox() */
    EIGEN_DEVICE_FUNC bool isApprox(const Transform& other, const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const
    {
        return m_matrix.isApprox(other.m_matrix, prec);
    }

    /** Sets the last row to [0 ... 0 1]
    */
    EIGEN_DEVICE_FUNC void makeAffine() { internal::transform_make_affine<int(Mode)>::run(m_matrix); }

    /** \internal
    * \returns the Dim x Dim linear part if the transformation is affine,
    *          and the HDim x Dim part for projective transformations.
    */
    EIGEN_DEVICE_FUNC inline Block<MatrixType, int(Mode) == int(Projective) ? HDim : Dim, Dim> linearExt()
    {
        return m_matrix.template block < int(Mode) == int(Projective) ? HDim : Dim, Dim > (0, 0);
    }
    /** \internal
    * \returns the Dim x Dim linear part if the transformation is affine,
    *          and the HDim x Dim part for projective transformations.
    */
    EIGEN_DEVICE_FUNC inline const Block<MatrixType, int(Mode) == int(Projective) ? HDim : Dim, Dim> linearExt() const
    {
        return m_matrix.template block < int(Mode) == int(Projective) ? HDim : Dim, Dim > (0, 0);
    }

    /** \internal
    * \returns the translation part if the transformation is affine,
    *          and the last column for projective transformations.
    */
    EIGEN_DEVICE_FUNC inline Block<MatrixType, int(Mode) == int(Projective) ? HDim : Dim, 1> translationExt()
    {
        return m_matrix.template block < int(Mode) == int(Projective) ? HDim : Dim, 1 > (0, Dim);
    }
    /** \internal
    * \returns the translation part if the transformation is affine,
    *          and the last column for projective transformations.
    */
    EIGEN_DEVICE_FUNC inline const Block<MatrixType, int(Mode) == int(Projective) ? HDim : Dim, 1> translationExt() const
    {
        return m_matrix.template block < int(Mode) == int(Projective) ? HDim : Dim, 1 > (0, Dim);
    }

#ifdef EIGEN_TRANSFORM_PLUGIN
#include EIGEN_TRANSFORM_PLUGIN
#endif

protected:
#ifndef EIGEN_PARSED_BY_DOXYGEN
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE void check_template_params()
    {
        EIGEN_STATIC_ASSERT((Options & (DontAlign | RowMajor)) == Options, INVALID_MATRIX_TEMPLATE_PARAMETERS)
    }
#endif
};

/** \ingroup Geometry_Module */
typedef Transform<float, 2, Isometry> Isometry2f;
/** \ingroup Geometry_Module */
typedef Transform<float, 3, Isometry> Isometry3f;
/** \ingroup Geometry_Module */
typedef Transform<double, 2, Isometry> Isometry2d;
/** \ingroup Geometry_Module */
typedef Transform<double, 3, Isometry> Isometry3d;

/** \ingroup Geometry_Module */
typedef Transform<float, 2, Affine> Affine2f;
/** \ingroup Geometry_Module */
typedef Transform<float, 3, Affine> Affine3f;
/** \ingroup Geometry_Module */
typedef Transform<double, 2, Affine> Affine2d;
/** \ingroup Geometry_Module */
typedef Transform<double, 3, Affine> Affine3d;

/** \ingroup Geometry_Module */
typedef Transform<float, 2, AffineCompact> AffineCompact2f;
/** \ingroup Geometry_Module */
typedef Transform<float, 3, AffineCompact> AffineCompact3f;
/** \ingroup Geometry_Module */
typedef Transform<double, 2, AffineCompact> AffineCompact2d;
/** \ingroup Geometry_Module */
typedef Transform<double, 3, AffineCompact> AffineCompact3d;

/** \ingroup Geometry_Module */
typedef Transform<float, 2, Projective> Projective2f;
/** \ingroup Geometry_Module */
typedef Transform<float, 3, Projective> Projective3f;
/** \ingroup Geometry_Module */
typedef Transform<double, 2, Projective> Projective2d;
/** \ingroup Geometry_Module */
typedef Transform<double, 3, Projective> Projective3d;

/**************************
*** Optional QT support ***
**************************/

#ifdef EIGEN_QT_SUPPORT
/** Initializes \c *this from a QMatrix assuming the dimension is 2.
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options> Transform<Scalar, Dim, Mode, Options>::Transform(const QMatrix& other)
{
    check_template_params();
    *this = other;
}

/** Set \c *this from a QMatrix assuming the dimension is 2.
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options>
Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::operator=(const QMatrix& other)
{
    EIGEN_STATIC_ASSERT(Dim == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
        m_matrix << other.m11(), other.m21(), other.dx(), other.m12(), other.m22(), other.dy();
    else
        m_matrix << other.m11(), other.m21(), other.dx(), other.m12(), other.m22(), other.dy(), 0, 0, 1;
    return *this;
}

/** \returns a QMatrix from \c *this assuming the dimension is 2.
  *
  * \warning this conversion might loss data if \c *this is not affine
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options> QMatrix Transform<Scalar, Dim, Mode, Options>::toQMatrix(void) const
{
    check_template_params();
    EIGEN_STATIC_ASSERT(Dim == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    return QMatrix(m_matrix.coeff(0, 0), m_matrix.coeff(1, 0), m_matrix.coeff(0, 1), m_matrix.coeff(1, 1), m_matrix.coeff(0, 2), m_matrix.coeff(1, 2));
}

/** Initializes \c *this from a QTransform assuming the dimension is 2.
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options> Transform<Scalar, Dim, Mode, Options>::Transform(const QTransform& other)
{
    check_template_params();
    *this = other;
}

/** Set \c *this from a QTransform assuming the dimension is 2.
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options>
Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::operator=(const QTransform& other)
{
    check_template_params();
    EIGEN_STATIC_ASSERT(Dim == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
        m_matrix << other.m11(), other.m21(), other.dx(), other.m12(), other.m22(), other.dy();
    else
        m_matrix << other.m11(), other.m21(), other.dx(), other.m12(), other.m22(), other.dy(), other.m13(), other.m23(), other.m33();
    return *this;
}

/** \returns a QTransform from \c *this assuming the dimension is 2.
  *
  * This function is available only if the token EIGEN_QT_SUPPORT is defined.
  */
template <typename Scalar, int Dim, int Mode, int Options> QTransform Transform<Scalar, Dim, Mode, Options>::toQTransform(void) const
{
    EIGEN_STATIC_ASSERT(Dim == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    if (EIGEN_CONST_CONDITIONAL(Mode == int(AffineCompact)))
        return QTransform(m_matrix.coeff(0, 0), m_matrix.coeff(1, 0), m_matrix.coeff(0, 1), m_matrix.coeff(1, 1), m_matrix.coeff(0, 2), m_matrix.coeff(1, 2));
    else
        return QTransform(m_matrix.coeff(0, 0),
                          m_matrix.coeff(1, 0),
                          m_matrix.coeff(2, 0),
                          m_matrix.coeff(0, 1),
                          m_matrix.coeff(1, 1),
                          m_matrix.coeff(2, 1),
                          m_matrix.coeff(0, 2),
                          m_matrix.coeff(1, 2),
                          m_matrix.coeff(2, 2));
}
#endif

/*********************
*** Procedural API ***
*********************/

/** Applies on the right the non uniform scale transformation represented
  * by the vector \a other to \c *this and returns a reference to \c *this.
  * \sa prescale()
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::scale(const MatrixBase<OtherDerived>& other)
{
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, int(Dim))
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    linearExt().noalias() = (linearExt() * other.asDiagonal());
    return *this;
}

/** Applies on the right a uniform scale of a factor \a c to \c *this
  * and returns a reference to \c *this.
  * \sa prescale(Scalar)
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::scale(const Scalar& s)
{
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    linearExt() *= s;
    return *this;
}

/** Applies on the left the non uniform scale transformation represented
  * by the vector \a other to \c *this and returns a reference to \c *this.
  * \sa scale()
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::prescale(const MatrixBase<OtherDerived>& other)
{
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, int(Dim))
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    affine().noalias() = (other.asDiagonal() * affine());
    return *this;
}

/** Applies on the left a uniform scale of a factor \a c to \c *this
  * and returns a reference to \c *this.
  * \sa scale(Scalar)
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::prescale(const Scalar& s)
{
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    m_matrix.template topRows<Dim>() *= s;
    return *this;
}

/** Applies on the right the translation matrix represented by the vector \a other
  * to \c *this and returns a reference to \c *this.
  * \sa pretranslate()
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::translate(const MatrixBase<OtherDerived>& other)
{
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, int(Dim))
    translationExt() += linearExt() * other;
    return *this;
}

/** Applies on the left the translation matrix represented by the vector \a other
  * to \c *this and returns a reference to \c *this.
  * \sa translate()
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::pretranslate(const MatrixBase<OtherDerived>& other)
{
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(OtherDerived, int(Dim))
    if (EIGEN_CONST_CONDITIONAL(int(Mode) == int(Projective)))
        affine() += other * m_matrix.row(Dim);
    else
        translation() += other;
    return *this;
}

/** Applies on the right the rotation represented by the rotation \a rotation
  * to \c *this and returns a reference to \c *this.
  *
  * The template parameter \a RotationType is the type of the rotation which
  * must be known by internal::toRotationMatrix<>.
  *
  * Natively supported types includes:
  *   - any scalar (2D),
  *   - a Dim x Dim matrix expression,
  *   - a Quaternion (3D),
  *   - a AngleAxis (3D)
  *
  * This mechanism is easily extendable to support user types such as Euler angles,
  * or a pair of Quaternion for 4D rotations.
  *
  * \sa rotate(Scalar), class Quaternion, class AngleAxis, prerotate(RotationType)
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename RotationType>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::rotate(const RotationType& rotation)
{
    linearExt() *= internal::toRotationMatrix<Scalar, Dim>(rotation);
    return *this;
}

/** Applies on the left the rotation represented by the rotation \a rotation
  * to \c *this and returns a reference to \c *this.
  *
  * See rotate() for further details.
  *
  * \sa rotate()
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename RotationType>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::prerotate(const RotationType& rotation)
{
    m_matrix.template block<Dim, HDim>(0, 0) = internal::toRotationMatrix<Scalar, Dim>(rotation) * m_matrix.template block<Dim, HDim>(0, 0);
    return *this;
}

/** Applies on the right the shear transformation represented
  * by the vector \a other to \c *this and returns a reference to \c *this.
  * \warning 2D only.
  * \sa preshear()
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::shear(const Scalar& sx, const Scalar& sy)
{
    EIGEN_STATIC_ASSERT(int(Dim) == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    VectorType tmp = linear().col(0) * sy + linear().col(1);
    linear() << linear().col(0) + linear().col(1) * sx, tmp;
    return *this;
}

/** Applies on the left the shear transformation represented
  * by the vector \a other to \c *this and returns a reference to \c *this.
  * \warning 2D only.
  * \sa shear()
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::preshear(const Scalar& sx, const Scalar& sy)
{
    EIGEN_STATIC_ASSERT(int(Dim) == 2, YOU_MADE_A_PROGRAMMING_MISTAKE)
    EIGEN_STATIC_ASSERT(Mode != int(Isometry), THIS_METHOD_IS_ONLY_FOR_SPECIFIC_TRANSFORMATIONS)
    m_matrix.template block<Dim, HDim>(0, 0) = LinearMatrixType(1, sx, sy, 1) * m_matrix.template block<Dim, HDim>(0, 0);
    return *this;
}

/******************************************************
*** Scaling, Translation and Rotation compatibility ***
******************************************************/

template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::operator=(const TranslationType& t)
{
    linear().setIdentity();
    translation() = t.vector();
    makeAffine();
    return *this;
}

template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options> Transform<Scalar, Dim, Mode, Options>::operator*(const TranslationType& t) const
{
    Transform res = *this;
    res.translate(t.vector());
    return res;
}

template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::operator=(const UniformScaling<Scalar>& s)
{
    m_matrix.setZero();
    linear().diagonal().fill(s.factor());
    makeAffine();
    return *this;
}

template <typename Scalar, int Dim, int Mode, int Options>
template <typename Derived>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options>& Transform<Scalar, Dim, Mode, Options>::operator=(const RotationBase<Derived, Dim>& r)
{
    linear() = internal::toRotationMatrix<Scalar, Dim>(r);
    translation().setZero();
    makeAffine();
    return *this;
}

template <typename Scalar, int Dim, int Mode, int Options>
template <typename Derived>
EIGEN_DEVICE_FUNC inline Transform<Scalar, Dim, Mode, Options> Transform<Scalar, Dim, Mode, Options>::operator*(const RotationBase<Derived, Dim>& r) const
{
    Transform res = *this;
    res.rotate(r.derived());
    return res;
}

/************************
*** Special functions ***
************************/

namespace internal {
    template <int Mode> struct transform_rotation_impl
    {
        template <typename TransformType> EIGEN_DEVICE_FUNC static inline const typename TransformType::LinearMatrixType run(const TransformType& t)
        {
            typedef typename TransformType::LinearMatrixType LinearMatrixType;
            LinearMatrixType result;
            t.computeRotationScaling(&result, (LinearMatrixType*)0);
            return result;
        }
    };
    template <> struct transform_rotation_impl<Isometry>
    {
        template <typename TransformType> EIGEN_DEVICE_FUNC static inline typename TransformType::ConstLinearPart run(const TransformType& t)
        {
            return t.linear();
        }
    };
}  // namespace internal
/** \returns the rotation part of the transformation
  *
  * If Mode==Isometry, then this method is an alias for linear(),
  * otherwise it calls computeRotationScaling() to extract the rotation
  * through a SVD decomposition.
  *
  * \svd_module
  *
  * \sa computeRotationScaling(), computeScalingRotation(), class SVD
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC typename Transform<Scalar, Dim, Mode, Options>::RotationReturnType Transform<Scalar, Dim, Mode, Options>::rotation() const
{
    return internal::transform_rotation_impl<Mode>::run(*this);
}

/** decomposes the linear part of the transformation as a product rotation x scaling, the scaling being
  * not necessarily positive.
  *
  * If either pointer is zero, the corresponding computation is skipped.
  *
  *
  *
  * \svd_module
  *
  * \sa computeScalingRotation(), rotation(), class SVD
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename RotationMatrixType, typename ScalingMatrixType>
EIGEN_DEVICE_FUNC void Transform<Scalar, Dim, Mode, Options>::computeRotationScaling(RotationMatrixType* rotation, ScalingMatrixType* scaling) const
{
    // Note that JacobiSVD is faster than BDCSVD for small matrices.
    JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);

    Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant() < Scalar(0) ? Scalar(-1) : Scalar(1);  // so x has absolute value 1
    VectorType sv(svd.singularValues());
    sv.coeffRef(Dim - 1) *= x;
    if (scaling)
        *scaling = svd.matrixV() * sv.asDiagonal() * svd.matrixV().adjoint();
    if (rotation)
    {
        LinearMatrixType m(svd.matrixU());
        m.col(Dim - 1) *= x;
        *rotation = m * svd.matrixV().adjoint();
    }
}

/** decomposes the linear part of the transformation as a product scaling x rotation, the scaling being
  * not necessarily positive.
  *
  * If either pointer is zero, the corresponding computation is skipped.
  *
  *
  *
  * \svd_module
  *
  * \sa computeRotationScaling(), rotation(), class SVD
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename ScalingMatrixType, typename RotationMatrixType>
EIGEN_DEVICE_FUNC void Transform<Scalar, Dim, Mode, Options>::computeScalingRotation(ScalingMatrixType* scaling, RotationMatrixType* rotation) const
{
    // Note that JacobiSVD is faster than BDCSVD for small matrices.
    JacobiSVD<LinearMatrixType> svd(linear(), ComputeFullU | ComputeFullV);

    Scalar x = (svd.matrixU() * svd.matrixV().adjoint()).determinant() < Scalar(0) ? Scalar(-1) : Scalar(1);  // so x has absolute value 1
    VectorType sv(svd.singularValues());
    sv.coeffRef(Dim - 1) *= x;
    if (scaling)
        *scaling = svd.matrixU() * sv.asDiagonal() * svd.matrixU().adjoint();
    if (rotation)
    {
        LinearMatrixType m(svd.matrixU());
        m.col(Dim - 1) *= x;
        *rotation = m * svd.matrixV().adjoint();
    }
}

/** Convenient method to set \c *this from a position, orientation and scale
  * of a 3D object.
  */
template <typename Scalar, int Dim, int Mode, int Options>
template <typename PositionDerived, typename OrientationType, typename ScaleDerived>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options>&
Transform<Scalar, Dim, Mode, Options>::fromPositionOrientationScale(const MatrixBase<PositionDerived>& position,
                                                                    const OrientationType& orientation,
                                                                    const MatrixBase<ScaleDerived>& scale)
{
    linear() = internal::toRotationMatrix<Scalar, Dim>(orientation);
    linear() *= scale.asDiagonal();
    translation() = position;
    makeAffine();
    return *this;
}

namespace internal {

    template <int Mode> struct transform_make_affine
    {
        template <typename MatrixType> EIGEN_DEVICE_FUNC static void run(MatrixType& mat)
        {
            static const int Dim = MatrixType::ColsAtCompileTime - 1;
            mat.template block<1, Dim>(Dim, 0).setZero();
            mat.coeffRef(Dim, Dim) = typename MatrixType::Scalar(1);
        }
    };

    template <> struct transform_make_affine<AffineCompact>
    {
        template <typename MatrixType> EIGEN_DEVICE_FUNC static void run(MatrixType&) {}
    };

    // selector needed to avoid taking the inverse of a 3x4 matrix
    template <typename TransformType, int Mode = TransformType::Mode> struct projective_transform_inverse
    {
        EIGEN_DEVICE_FUNC static inline void run(const TransformType&, TransformType&) {}
    };

    template <typename TransformType> struct projective_transform_inverse<TransformType, Projective>
    {
        EIGEN_DEVICE_FUNC static inline void run(const TransformType& m, TransformType& res) { res.matrix() = m.matrix().inverse(); }
    };

}  // end namespace internal

/**
  *
  * \returns the inverse transformation according to some given knowledge
  * on \c *this.
  *
  * \param hint allows to optimize the inversion process when the transformation
  * is known to be not a general transformation (optional). The possible values are:
  *  - #Projective if the transformation is not necessarily affine, i.e., if the
  *    last row is not guaranteed to be [0 ... 0 1]
  *  - #Affine if the last row can be assumed to be [0 ... 0 1]
  *  - #Isometry if the transformation is only a concatenations of translations
  *    and rotations.
  *  The default is the template class parameter \c Mode.
  *
  * \warning unless \a traits is always set to NoShear or NoScaling, this function
  * requires the generic inverse method of MatrixBase defined in the LU module. If
  * you forget to include this module, then you will get hard to debug linking errors.
  *
  * \sa MatrixBase::inverse()
  */
template <typename Scalar, int Dim, int Mode, int Options>
EIGEN_DEVICE_FUNC Transform<Scalar, Dim, Mode, Options> Transform<Scalar, Dim, Mode, Options>::inverse(TransformTraits hint) const
{
    Transform res;
    if (hint == Projective)
    {
        internal::projective_transform_inverse<Transform>::run(*this, res);
    }
    else
    {
        if (hint == Isometry)
        {
            res.matrix().template topLeftCorner<Dim, Dim>() = linear().transpose();
        }
        else if (hint & Affine)
        {
            res.matrix().template topLeftCorner<Dim, Dim>() = linear().inverse();
        }
        else
        {
            eigen_assert(false && "Invalid transform traits in Transform::Inverse");
        }
        // translation and remaining parts
        res.matrix().template topRightCorner<Dim, 1>() = -res.matrix().template topLeftCorner<Dim, Dim>() * translation();
        res.makeAffine();  // we do need this, because in the beginning res is uninitialized
    }
    return res;
}

namespace internal {

    /*****************************************************
*** Specializations of take affine part            ***
*****************************************************/

    template <typename TransformType> struct transform_take_affine_part
    {
        typedef typename TransformType::MatrixType MatrixType;
        typedef typename TransformType::AffinePart AffinePart;
        typedef typename TransformType::ConstAffinePart ConstAffinePart;
        static inline AffinePart run(MatrixType& m) { return m.template block<TransformType::Dim, TransformType::HDim>(0, 0); }
        static inline ConstAffinePart run(const MatrixType& m) { return m.template block<TransformType::Dim, TransformType::HDim>(0, 0); }
    };

    template <typename Scalar, int Dim, int Options> struct transform_take_affine_part<Transform<Scalar, Dim, AffineCompact, Options>>
    {
        typedef typename Transform<Scalar, Dim, AffineCompact, Options>::MatrixType MatrixType;
        static inline MatrixType& run(MatrixType& m) { return m; }
        static inline const MatrixType& run(const MatrixType& m) { return m; }
    };

    /*****************************************************
*** Specializations of construct from matrix       ***
*****************************************************/

    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_construct_from_matrix<Other, Mode, Options, Dim, HDim, Dim, Dim>
    {
        static inline void run(Transform<typename Other::Scalar, Dim, Mode, Options>* transform, const Other& other)
        {
            transform->linear() = other;
            transform->translation().setZero();
            transform->makeAffine();
        }
    };

    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_construct_from_matrix<Other, Mode, Options, Dim, HDim, Dim, HDim>
    {
        static inline void run(Transform<typename Other::Scalar, Dim, Mode, Options>* transform, const Other& other)
        {
            transform->affine() = other;
            transform->makeAffine();
        }
    };

    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_construct_from_matrix<Other, Mode, Options, Dim, HDim, HDim, HDim>
    {
        static inline void run(Transform<typename Other::Scalar, Dim, Mode, Options>* transform, const Other& other) { transform->matrix() = other; }
    };

    template <typename Other, int Options, int Dim, int HDim> struct transform_construct_from_matrix<Other, AffineCompact, Options, Dim, HDim, HDim, HDim>
    {
        static inline void run(Transform<typename Other::Scalar, Dim, AffineCompact, Options>* transform, const Other& other)
        {
            transform->matrix() = other.template block<Dim, HDim>(0, 0);
        }
    };

    /**********************************************************
***   Specializations of operator* with rhs EigenBase   ***
**********************************************************/

    template <int LhsMode, int RhsMode> struct transform_product_result
    {
        enum
        {
            Mode = (LhsMode == (int)Projective || RhsMode == (int)Projective) ?
                       Projective :
                       (LhsMode == (int)Affine || RhsMode == (int)Affine) ? Affine :
                                                                            (LhsMode == (int)AffineCompact || RhsMode == (int)AffineCompact) ?
                                                                            AffineCompact :
                                                                            (LhsMode == (int)Isometry || RhsMode == (int)Isometry) ? Isometry : Projective
        };
    };

    template <typename TransformType, typename MatrixType, int RhsCols> struct transform_right_product_impl<TransformType, MatrixType, 0, RhsCols>
    {
        typedef typename MatrixType::PlainObject ResultType;

        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other) { return T.matrix() * other; }
    };

    template <typename TransformType, typename MatrixType, int RhsCols> struct transform_right_product_impl<TransformType, MatrixType, 1, RhsCols>
    {
        enum
        {
            Dim = TransformType::Dim,
            HDim = TransformType::HDim,
            OtherRows = MatrixType::RowsAtCompileTime,
            OtherCols = MatrixType::ColsAtCompileTime
        };

        typedef typename MatrixType::PlainObject ResultType;

        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
        {
            EIGEN_STATIC_ASSERT(OtherRows == HDim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);

            typedef Block<ResultType, Dim, OtherCols, int(MatrixType::RowsAtCompileTime) == Dim> TopLeftLhs;

            ResultType res(other.rows(), other.cols());
            TopLeftLhs(res, 0, 0, Dim, other.cols()).noalias() = T.affine() * other;
            res.row(OtherRows - 1) = other.row(OtherRows - 1);

            return res;
        }
    };

    template <typename TransformType, typename MatrixType, int RhsCols> struct transform_right_product_impl<TransformType, MatrixType, 2, RhsCols>
    {
        enum
        {
            Dim = TransformType::Dim,
            HDim = TransformType::HDim,
            OtherRows = MatrixType::RowsAtCompileTime,
            OtherCols = MatrixType::ColsAtCompileTime
        };

        typedef typename MatrixType::PlainObject ResultType;

        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
        {
            EIGEN_STATIC_ASSERT(OtherRows == Dim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);

            typedef Block<ResultType, Dim, OtherCols, true> TopLeftLhs;
            ResultType res(Replicate<typename TransformType::ConstTranslationPart, 1, OtherCols>(T.translation(), 1, other.cols()));
            TopLeftLhs(res, 0, 0, Dim, other.cols()).noalias() += T.linear() * other;

            return res;
        }
    };

    template <typename TransformType, typename MatrixType> struct transform_right_product_impl<TransformType, MatrixType, 2, 1>  // rhs is a vector of size Dim
    {
        typedef typename TransformType::MatrixType TransformMatrix;
        enum
        {
            Dim = TransformType::Dim,
            HDim = TransformType::HDim,
            OtherRows = MatrixType::RowsAtCompileTime,
            WorkingRows = EIGEN_PLAIN_ENUM_MIN(TransformMatrix::RowsAtCompileTime, HDim)
        };

        typedef typename MatrixType::PlainObject ResultType;

        static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ResultType run(const TransformType& T, const MatrixType& other)
        {
            EIGEN_STATIC_ASSERT(OtherRows == Dim, YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES);

            Matrix<typename ResultType::Scalar, Dim + 1, 1> rhs;
            rhs.template head<Dim>() = other;
            rhs[Dim] = typename ResultType::Scalar(1);
            Matrix<typename ResultType::Scalar, WorkingRows, 1> res(T.matrix() * rhs);
            return res.template head<Dim>();
        }
    };

    /**********************************************************
***   Specializations of operator* with lhs EigenBase   ***
**********************************************************/

    // generic HDim x HDim matrix * T => Projective
    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_left_product_impl<Other, Mode, Options, Dim, HDim, HDim, HDim>
    {
        typedef Transform<typename Other::Scalar, Dim, Mode, Options> TransformType;
        typedef typename TransformType::MatrixType MatrixType;
        typedef Transform<typename Other::Scalar, Dim, Projective, Options> ResultType;
        static ResultType run(const Other& other, const TransformType& tr) { return ResultType(other * tr.matrix()); }
    };

    // generic HDim x HDim matrix * AffineCompact => Projective
    template <typename Other, int Options, int Dim, int HDim> struct transform_left_product_impl<Other, AffineCompact, Options, Dim, HDim, HDim, HDim>
    {
        typedef Transform<typename Other::Scalar, Dim, AffineCompact, Options> TransformType;
        typedef typename TransformType::MatrixType MatrixType;
        typedef Transform<typename Other::Scalar, Dim, Projective, Options> ResultType;
        static ResultType run(const Other& other, const TransformType& tr)
        {
            ResultType res;
            res.matrix().noalias() = other.template block<HDim, Dim>(0, 0) * tr.matrix();
            res.matrix().col(Dim) += other.col(Dim);
            return res;
        }
    };

    // affine matrix * T
    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_left_product_impl<Other, Mode, Options, Dim, HDim, Dim, HDim>
    {
        typedef Transform<typename Other::Scalar, Dim, Mode, Options> TransformType;
        typedef typename TransformType::MatrixType MatrixType;
        typedef TransformType ResultType;
        static ResultType run(const Other& other, const TransformType& tr)
        {
            ResultType res;
            res.affine().noalias() = other * tr.matrix();
            res.matrix().row(Dim) = tr.matrix().row(Dim);
            return res;
        }
    };

    // affine matrix * AffineCompact
    template <typename Other, int Options, int Dim, int HDim> struct transform_left_product_impl<Other, AffineCompact, Options, Dim, HDim, Dim, HDim>
    {
        typedef Transform<typename Other::Scalar, Dim, AffineCompact, Options> TransformType;
        typedef typename TransformType::MatrixType MatrixType;
        typedef TransformType ResultType;
        static ResultType run(const Other& other, const TransformType& tr)
        {
            ResultType res;
            res.matrix().noalias() = other.template block<Dim, Dim>(0, 0) * tr.matrix();
            res.translation() += other.col(Dim);
            return res;
        }
    };

    // linear matrix * T
    template <typename Other, int Mode, int Options, int Dim, int HDim> struct transform_left_product_impl<Other, Mode, Options, Dim, HDim, Dim, Dim>
    {
        typedef Transform<typename Other::Scalar, Dim, Mode, Options> TransformType;
        typedef typename TransformType::MatrixType MatrixType;
        typedef TransformType ResultType;
        static ResultType run(const Other& other, const TransformType& tr)
        {
            TransformType res;
            if (Mode != int(AffineCompact))
                res.matrix().row(Dim) = tr.matrix().row(Dim);
            res.matrix().template topRows<Dim>().noalias() = other * tr.matrix().template topRows<Dim>();
            return res;
        }
    };

    /**********************************************************
*** Specializations of operator* with another Transform ***
**********************************************************/

    template <typename Scalar, int Dim, int LhsMode, int LhsOptions, int RhsMode, int RhsOptions>
    struct transform_transform_product_impl<Transform<Scalar, Dim, LhsMode, LhsOptions>, Transform<Scalar, Dim, RhsMode, RhsOptions>, false>
    {
        enum
        {
            ResultMode = transform_product_result<LhsMode, RhsMode>::Mode
        };
        typedef Transform<Scalar, Dim, LhsMode, LhsOptions> Lhs;
        typedef Transform<Scalar, Dim, RhsMode, RhsOptions> Rhs;
        typedef Transform<Scalar, Dim, ResultMode, LhsOptions> ResultType;
        static ResultType run(const Lhs& lhs, const Rhs& rhs)
        {
            ResultType res;
            res.linear() = lhs.linear() * rhs.linear();
            res.translation() = lhs.linear() * rhs.translation() + lhs.translation();
            res.makeAffine();
            return res;
        }
    };

    template <typename Scalar, int Dim, int LhsMode, int LhsOptions, int RhsMode, int RhsOptions>
    struct transform_transform_product_impl<Transform<Scalar, Dim, LhsMode, LhsOptions>, Transform<Scalar, Dim, RhsMode, RhsOptions>, true>
    {
        typedef Transform<Scalar, Dim, LhsMode, LhsOptions> Lhs;
        typedef Transform<Scalar, Dim, RhsMode, RhsOptions> Rhs;
        typedef Transform<Scalar, Dim, Projective> ResultType;
        static ResultType run(const Lhs& lhs, const Rhs& rhs) { return ResultType(lhs.matrix() * rhs.matrix()); }
    };

    template <typename Scalar, int Dim, int LhsOptions, int RhsOptions>
    struct transform_transform_product_impl<Transform<Scalar, Dim, AffineCompact, LhsOptions>, Transform<Scalar, Dim, Projective, RhsOptions>, true>
    {
        typedef Transform<Scalar, Dim, AffineCompact, LhsOptions> Lhs;
        typedef Transform<Scalar, Dim, Projective, RhsOptions> Rhs;
        typedef Transform<Scalar, Dim, Projective> ResultType;
        static ResultType run(const Lhs& lhs, const Rhs& rhs)
        {
            ResultType res;
            res.matrix().template topRows<Dim>() = lhs.matrix() * rhs.matrix();
            res.matrix().row(Dim) = rhs.matrix().row(Dim);
            return res;
        }
    };

    template <typename Scalar, int Dim, int LhsOptions, int RhsOptions>
    struct transform_transform_product_impl<Transform<Scalar, Dim, Projective, LhsOptions>, Transform<Scalar, Dim, AffineCompact, RhsOptions>, true>
    {
        typedef Transform<Scalar, Dim, Projective, LhsOptions> Lhs;
        typedef Transform<Scalar, Dim, AffineCompact, RhsOptions> Rhs;
        typedef Transform<Scalar, Dim, Projective> ResultType;
        static ResultType run(const Lhs& lhs, const Rhs& rhs)
        {
            ResultType res(lhs.matrix().template leftCols<Dim>() * rhs.matrix());
            res.matrix().col(Dim) += lhs.matrix().col(Dim);
            return res;
        }
    };

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_TRANSFORM_H
